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A note on the limit points associated with the generalized \(abc\)-conjecture for \(\mathbb{Z} [t]\). (English) Zbl 0880.11026
This paper constructs examples to show that a conjectural generalization of the \(abc\) conjecture to \(n\) variables is best possible over function fields. Let \(K\) be an integral domain of characteristic \(0\). For nonzero polynomials \(A(t)\in K[t]\) the radical of \(A\), denoted \(\text{rad}(A)\), is the product of all distinct irreducible factors of \(A\). For fixed \(n\geq3\) let \(T_n\) denote the set of \(n\)-tuples \((A_1(t),\dots,A_n(t))\in K[t]^n\) such that \(A_1+\dots+A_n=0\), such that no subsum of the \(A_i\) equals zero, and such that \(\gcd(A_1,\dots,A_n)=1\). The paper constructs examples showing that the limit set of the ratio \[ \frac{\max\{\deg A_1,\dots,\deg A_n\}} {\deg(\text{rad}(A_1\cdot\dots\cdot A_n))} , \] as \((A_1,\dots,A_n)\) varies over \(T_n\), contains the interval \([1/n,2n-5]\). The Masser-Oesterlé \(abc\) conjecture asserts that, if \(n=3\) and if \(K[t]\) is replaced by \(\mathbb Z\), then the lim sup of the corresponding ratio is \(\leq 1\). This conjecture was generalized to arbitrary \(n\geq3\) by J. Browkin and J. Brzeziński [Math. Comput. 62, 931-939 (1994; Zbl 0804.11006)], who showed that the lim sup is \(\geq 2n-5\) over number fields, and conjectured that the lim sup was equal to \(2n-5\).
Reviewer: P.Vojta (Berkeley)
11C08 Polynomials in number theory
11D04 Linear Diophantine equations
11A99 Elementary number theory
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